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Overpartitions related to the mock theta function $\omega (q)$

Volume 181 / 2017

George E. Andrews, Atul Dixit, Daniel Schultz, Ae Ja Yee Acta Arithmetica 181 (2017), 253-286 MSC: Primary 11P81; Secondary 05A17. DOI: 10.4064/aa161225-7-11 Published online: 1 December 2017

Abstract

It was recently shown that $q\omega (q)$, where $\omega (q)$ is one of the third order mock theta functions, is the generating function of $p_{\omega }(n)$, the number of partitions of a positive integer $n$ such that all odd parts are less than twice the smallest part. In this paper, we study the overpartition analogue of $p_{\omega }(n)$, and express its generating function in terms of a ${}_3\phi _{2}$ basic hypergeometric series and an infinite series involving little $q$-Jacobi polynomials. This is accomplished by obtaining a new seven-parameter $q$-series identity which generalizes a deep identity due to the first author as well as its generalization by R. P. Agarwal. We also derive two interesting congruences satisfied by the overpartition analogue, and some congruences satisfied by the associated smallest parts function.

Authors

  • George E. AndrewsDepartment of Mathematics
    The Pennsylvania State University
    University Park, PA 16802, U.S.A.
    e-mail
  • Atul DixitDepartment of Mathematics
    Indian Institute of Technology Gandhinagar
    Palaj, Gandhinagar, Gujarat 382355, India
    e-mail
  • Daniel SchultzDepartment of Mathematics
    The Pennsylvania State University
    University Park, PA 16802, U.S.A.
    e-mail
  • Ae Ja YeeDepartment of Mathematics
    The Pennsylvania State University
    University Park, PA 16802, U.S.A.
    e-mail

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